Standard Deviation Calculator
Sample and population spread, variance and standard error.
Reviewed by the Calculation Machine team · Updated 24 October 2026 · How we check our maths
Your values
ⓘ Live results
- Sample standard deviation
- 2.160247
- Population standard deviation
- 2
- Sample variance
- 4.666667
- Mean
- 6
- Standard error of the mean
- 0.816497
Working
- 1n = 7, Σ(x − x̄)² = 28
Quick answer
Standard deviation measures how spread out numbers are around their mean. Subtract the mean from each value, square the differences, add them, divide by n − 1 for a sample (or n for a whole population), then take the square root. For 4, 8, 6, 5, 3, 9, 7 the mean is 6, the squared differences sum to 28, and the sample SD is √(28 ÷ 6) = 2.16.
What the Standard Deviation calculator does
Paste or type a list of numbers separated by commas and this calculator returns the sample standard deviation, the population standard deviation, the sample variance, the mean and the standard error of the mean (SEM). The working line shows n and the sum of squared deviations so you can check a hand calculation.
Use the sample figure when your numbers are a subset drawn from something larger – survey responses, a batch of measured parts, a few weeks of sales – which is almost always the case. Use the population figure only when you have every value in the group, such as the scores of all students in one class when that class is all you care about.
Standard deviation is the basis for z-scores, confidence intervals and the empirical rule. Enter at least two values; blank entries and extra spaces are ignored.
The formula
Sample SD: s = √( Σ(xᵢ − x̄)² ÷ (n − 1) ) Population SD: σ = √( Σ(xᵢ − μ)² ÷ n ) Standard error of the mean: SEM = s ÷ √n
| Symbol | Meaning | Unit |
|---|---|---|
| xᵢ | Each individual value in the data set | same as data |
| x̄ or μ | Mean of the values (sample mean or population mean) | same as data |
| n | Number of values | count |
| n − 1 | Degrees of freedom; Bessel's correction for estimating spread from a sample | count |
These are the standard definitions found in any introductory statistics text and in the NIST/SEMATECH e-Handbook of Statistical Methods. Dividing by n − 1 (Bessel's correction) stops a sample from understating the spread of the population it came from. Variance is the SD squared and is in squared units. The calculator treats values as raw data; for grouped frequency tables, expand the values or use a weighted method.
Worked examples
Exam scores for a class section
A teacher enters eight test scores out of 100 to see how consistent the class was.
Inputs
- Data (comma separated)72, 85, 90, 66, 78, 88, 95, 81
Result
- Sample standard deviation9.642725
- Population standard deviation9.019943
- Sample variance92.982143
- Mean81.875
- Standard error of the mean3.409218
The mean is about 81.9 with a sample SD of about 9.6 points. Roughly two-thirds of similar students would be expected within one SD of the mean, between about 72 and 92, if scores are near-normal.
Precision of machined parts
A quality inspector measures six 10 mm pins: 9.98, 10.02, 10.01, 9.97, 10.03 and 10.00 mm.
Inputs
- Data (comma separated)9.98, 10.02, 10.01, 9.97, 10.03, 10.00
Result
- Sample standard deviation0.023166
- Population standard deviation0.021148
- Sample variance0.000537
- Mean10.001667
- Standard error of the mean0.009458
The sample SD is about 0.023 mm. With a ±0.05 mm tolerance, the process sits a little over two SDs from each limit – workable, but a larger sample would be needed to judge process capability.
Weekly sales with one outlier
A shop owner enters seven weeks of sales in dollars, including one holiday week of $2,600.
Inputs
- Data (comma separated)1200, 950, 1430, 1100, 2600, 1050, 1180
Result
- Sample standard deviation567.492396
- Population standard deviation525.395866
- Sample variance322047.619048
- Mean1358.571429
- Standard error of the mean214.491964
The sample SD is about $567. Remove the holiday week and it drops to about $164, showing how a single outlier inflates standard deviation. Check for outliers with the outlier calculator before reporting spread.
Sample or population: which to report
The question is whether you want to describe only the numbers in front of you or use them to say something about a larger group. If it is the second – and it usually is – use the sample figure. With large n the two converge: at n = 100 they differ by about 0.5%.
| Situation | Use | Spreadsheet function |
|---|---|---|
| Survey of 200 customers out of thousands | Sample (n − 1) | STDEV.S |
| Lab measurements of a few specimens | Sample (n − 1) | STDEV.S |
| Every employee's salary in a 12-person company, describing only that company | Population (n) | STDEV.P |
| All 30 days of a month's data, used to predict future months | Sample (n − 1) | STDEV.S |
Standard error of the mean and root mean square
Standard error of the mean (SEM) is the sample SD divided by √n. It measures how precisely the sample mean estimates the true mean, not how spread out the data are. For the default data (sample SD 2.16, n = 7), SEM = 2.16 ÷ √7 ≈ 0.82. SEM shrinks as you collect more data; SD does not. A 95% confidence interval for the mean is roughly the mean ± 2 SEM for moderate samples.
Root mean square (RMS) is √(Σx² ÷ n) – the square root of the average of the squared values, used for AC voltages, signal levels and error measures. It is linked to the outputs here: RMS² = mean² + population SD². For 4, 8, 6, 5, 3, 9, 7 that gives √(6² + 2²) = √40 ≈ 6.32. When the mean is zero, as with a pure AC signal or a set of residuals, RMS equals the population SD.
Reading the result
- Same units as the data: an SD of 9.6 points on a test means typical scores sit about 9.6 points from the average.
- Relative spread: to compare spread across data sets with different means, divide SD by the mean – the coefficient of variation.
- Normal data: about 68% of values fall within 1 SD of the mean, 95% within 2 SD and 99.7% within 3 SD.
- Skewed data: incomes, house prices and waiting times are skewed, so SD can mislead; the interquartile range is often a better summary.
How to use the Standard Deviation Calculator
- 1Paste or type your values separated by commas, for example from a spreadsheet column.
- 2Check that units match – do not mix centimeters and inches in one list.
- 3Read the sample SD if your data are a subset, or the population SD if they are the complete group.
- 4Use the mean and standard error together to report how precise an average is.
Frequently asked questions
What is the difference between sample and population standard deviation?
Population SD divides the sum of squared deviations by n; sample SD divides by n − 1. The sample version is slightly larger and corrects for the tendency of a sample to underestimate spread.
What is a good standard deviation?
There is no universal good value; it depends on the data. In manufacturing a small SD relative to the tolerance is good, while in investing a higher SD means more volatile returns.
How is variance related to standard deviation?
Variance is the standard deviation squared. A sample SD of 2.16 corresponds to a variance of about 4.67, in squared units of the original data.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is zero when all values are identical and positive otherwise.
Why does the calculator need at least two values?
A single value has no spread, and the sample formula would divide by n − 1 = 0, which is undefined.
References
- NIST/SEMATECH e-Handbook of Statistical Methods — National Institute of Standards and Technology
- OpenIntro Statistics — OpenIntro
About this calculator
- Built and reviewed by the Calculation Machine team · How we build and test calculators
- Last reviewed: October 24, 2026
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